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Properties

Label 2-31e2-31.23-c0-0-0
Degree $2$
Conductor $961$
Sign $-0.962 - 0.272i$
Analytic cond. $0.479601$
Root an. cond. $0.692532$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 + 0.951i)2-s − 5-s + (0.809 + 0.587i)7-s + (−0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + (0.309 − 0.951i)10-s + (−0.809 + 0.587i)14-s + (−0.309 − 0.951i)16-s + (−0.309 − 0.951i)18-s + (−0.309 + 0.951i)19-s + (−0.809 − 0.587i)35-s + (−0.809 − 0.587i)38-s + (0.809 − 0.587i)40-s + (−0.309 + 0.951i)41-s + (0.809 − 0.587i)45-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s − 5-s + (0.809 + 0.587i)7-s + (−0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + (0.309 − 0.951i)10-s + (−0.809 + 0.587i)14-s + (−0.309 − 0.951i)16-s + (−0.309 − 0.951i)18-s + (−0.309 + 0.951i)19-s + (−0.809 − 0.587i)35-s + (−0.809 − 0.587i)38-s + (0.809 − 0.587i)40-s + (−0.309 + 0.951i)41-s + (0.809 − 0.587i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $-0.962 - 0.272i$
Analytic conductor: \(0.479601\)
Root analytic conductor: \(0.692532\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 961,\ (\ :0),\ -0.962 - 0.272i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6560894341\)
\(L(\frac12)\) \(\approx\) \(0.6560894341\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
good2 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
3 \( 1 + (0.809 - 0.587i)T^{2} \)
5 \( 1 + T + T^{2} \)
7 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
11 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + (0.809 + 0.587i)T^{2} \)
47 \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 2T + T^{2} \)
71 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 + (-0.309 - 0.951i)T^{2} \)
97 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.81522494485060276984392642421, −9.457035726204964027110863817371, −8.438107965760562356600835091255, −8.099496555798474236298108412515, −7.54894523631049472777080861732, −6.39132316265927062708174563964, −5.58962474294699969164112923343, −4.70309477190647836673756099552, −3.38262884136449105992825694290, −2.18994136627161153429654453173, 0.67741350491287073570256142602, 2.22380342541582270248557549973, 3.40132014597364513401999463594, 4.15823935514117878179352947406, 5.38459402420678534271156286873, 6.58543501692979862242916410041, 7.39745140265450717735596783736, 8.414329633425062275956077429185, 9.010486739525916978806027062674, 10.06633502658368932509387396743

Graph of the $Z$-function along the critical line